Half-spherical twists on derived categories of coherent sheaves

2023/02/24 by Arai, Hayato
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2302.12501

Abstract

For a flat morphism π\colon X → T between smooth quasi-projective varieties and its fiber X₀, we prove that spherical objects on D^b(X) pushed-forward from D^b(X₀) induce autoequivalences of D^b(X₀) itself. Our construction provides new derived symmetries for some singular varieties, which include singular fibers of elliptic surfaces (commonly referred to as Kodaira fibers) and type III degenerations of K3 surfaces. In the case of Kodaira fibers of type I_n, we also show the induced autoequivalences of D^b(X₀) correspond to the half twists on the n-punctured 2-torus via homological mirror symmetry. As an application, we describe the autoequivalence groups of elliptic surfaces in terms of mapping class groups of punctured tori.

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